Ancient Mathematics - the slides will show you how ancient mathematics developed and flourished

1. Ancient Mathematics

2. Egyptian Geometry:
Approximation

3. Approximating the Area of a
Quadrilateral
• Egyptian Formula
• A = 1.4 (a+c)(b+d)
• Where a,b,c,d are the lengths of the
consecutive sides
• Is the formula correct? When will this be
correct? Prove!

4. Approximating the area of a circle
• Example of Egyptian problems: A round field
of diameter 9 khet. What is its area?
• Solution: Take away 1/9 of the diameter,
namely 1; the remainder is 8; it makes 64.
Therefore it contains 64 setat of land.
• The formula here is A = (8d/9)^2.
• If this was the formula, what was the
approximate value of ∏, pi.

5. Babylonian: Approximating the value
of ∏
• The circumference of a circle was found by
taking three tomes its diameter. What was
their value for pi?

6. Approximating the area of a trapezoid
• A = ½ (b + b’)h
• b and b’ are the parallel sides; h is the height.
• Is this formula correct?

7. Approximating the Volume of a
Truncated Pyramid
• V = h/3 ( a^2 + ab + b^2)
• Where h is the height and a and b are the
parallel side
• Is this correct? Compare it the formula V = h/3
( B + b + (√Bb))

8. Approximating the area of a square
prism
• V = h/3 x a^2
• Is this correct? Prove

9. Word Problems
• Refer to the examples to be given (pp.61-63)

10. Babylonian Mathematics: A tablet of
reciprocals
• A pair of numbers whose product is 60
4 15
5 12
6 10
8 7;30
9 6;40
10 6
12 5
16 3;45
18 3;20

11. Analysis
• 8 7; 30
• -the left side is the quotient; the right is the
reciprocal based on the sexagesimal system

12. Complete the table
• Rule: product must be 50
12
4
7
8
9
11
13

13. The Babylonian Treatment of
Quadratic Equations
• Their formula was
• Prove the formula

14. Plimpton 322: A tablet concerning
number triplets
• X^2 + y^2 = Z^2 (similar to a^2 + b^2 = c^2)
• Z^2 – y^2 = x^2 ( divide by x)
• If α = z/x and β= y/x
• α ^2 – β^2 =1 (factor out)
• 1 to be represented as (m/n)(n/m)
• Apply APE, α = ½ (m/n + n/m); β = ½ ( m/n – n/m)
• The formula are α = (m^2 + n^2 )/2mn; β = (m^2-
n^2)/2mn
• But y = β x, z = α x and x = 2mn
• X= 2mn, y = m^2-n^2; z = m^2 + n^2

15. Verify the formula: X= 2mn, y = m^2-
n^2; z = m^2 + n^2

16. The Cairo Mathematical Papyrus
• Solve x^2 + y^2 = 169; xy = 60